This invention relates in general to a system and method for nonlinear frequency conversion tunable laser light using achromatic phase-matching, and in particular to an achromatic phase-matching optical system and method which exactly matches the high order dispersion characteristics of nonlinear optical materials.
Many applications require broadly tunable UV light. No such laser source exists, however, so tunable UV is usually obtained by frequency-doubling a tunable laser in the visible and near-IR by using nonlinear optical effects such as a second harmonic generation process. Such processes are phenomenon which derive from nonlinear polarization effects of certain material media. The effect depends upon crystal structure, particularly anisotropic structure. Commonly used crystals are .beta.-barium borate ("BBO"), potassium dihydrogen phosphate ("KDP") and lithium triborate ("LBO").
Because frequency-doubling, therefore, involves passing light through a nonlinear crystal, and the effects of its wavelength-dependent refractive index must be taken into account. In particular, in order for frequency doubling to take place in the crystal, the refractive index of the incident light alt the "fundamental" wavelength must equal the refractive index of the frequency doubled light to be produced. Since the refractive index of the crystal varies botlr with the angle of incidence and with the frequency of the input beam it is apparent the that absent extraordinary precaution only a very narrow range of frequencies of a broadband beam can enter a crystal at the appropriate incident angle for efficient frequency doubling. Unfortunately, this procedure is sensitive to vibrations and can be unreliable, despite the use of feedback. In addition, it produces undesirable beam walk as the laser tunes, which must be corrected with yet another moving part.
Second-harmonic generation of light (hereafter referred to as "SHG") the generation of light of twice the optical frequency of input laser light, has been an essential tool of laser research for many years. It is used widely to generate ultraviolet light because such wavelengths are difficult to generate directly from a laser. Indeed, this technique is often used to generate visible light from a near-infrared laser because it is easier to generate near-infrared laser light than it is to generate visible light. In general, however, it is possible to frequency-double light from virtually all visible and near-infrared lasers.
A particular type of laser light which is important to frequency-double is broadband light. However, the use of SHG processes to frequency-double broadband light which is incoherent has proved to be difficult and inefficient. (In general, ultrashort pulses generated by lasers can be considered broadband light whose frequencies are in phase while incoherent light can be considered broadband light whose frequencies are randomly phased.) These two types of light are difficult to frequency-double due to their respective large bandwidths. As a result of the large bandwidths, efficient methods for frequency-doubling both of these types of light have not been developed.
The efficiency .eta. of a SHG process depends on several factors. A first factor is the nonlinear coefficient of a SHG crystal used. This factor depends on internal properties of the crystal and can only be improved by manipulating the composition of the crystal.
Second, .eta. is proportional to the square of the length of the crystal, L, the distant through which light ray propagate through the crystal. Thus, thick crystals yield much higher efficiency than thin ones.
Third, .eta. depends on the laser intensity and is, typically, directly proportional to the laser intensity. Consequently, continuous-beam lasers, which have relatively low intensity, frequency-double inefficiently while pulsed lasers, which generally achieve higher intensity, frequency-double more efficiently. In general, the shorter the pulse the more efficiently it frequency-doubles, given a fixed energy per pulse.
As earlier noted, in order for frequency-doubling to take place in a SHG crystals, the refractive index of the input laser light (again, the "fundamental" wavelength) must equal the refractive index of the frequency-doubled light to be produced. Since the refractive index of a crystal is a function of both the incidence angle and frequency of the input beam different incidence angles must be used to obtain maximum efficiency .eta. for different wavelengths. The requirement that a wavelength enter the crystal at the appropriate angle necessary to frequency-double most efficiently will be referred to hereinafter as the "phase-matching condition," or simply "phase-matching" for short. The angle will be referred to as the "phase-matching angle," and is a function of wavelength.
Because the efficiency .eta. of the SHG process is strongly "peaked" with respect to the entrance angle for a given wavelength and also with respect to wavelength for a given angle only a small very narrow range of wavelengths near the exact phase-matching wavelength can still yield highly efficient SHG process. The range of wavelengths that achieves high-efficiency frequency-doubling for a single angle is called the crystal's "phase-matching bandwidth" for that angle. If the input laser light contains frequencies outside this bandwidth, such frequencies will not produce their corresponding second harmonic (i.e., will not be frequency-doubled and the efficiency of the overall process is reduced.
When the crystal bandwidth is greater than the input light bandwidth, the above effect can be neglected. However, when the crystal bandwidth is less than the bandwidth of the input light, the SHG efficiency is proportional to the crystal bandwidth, yielding a fourth factor. In this case, the efficiency can be written approximately as: ##EQU1##
where d is the nonlinear coefficient of the crystal, I is the intensity of the light, L is the length of the crystal through which the light propagates, .DELTA..lambda..sub.cr is the bandwidth of the crystal, and .DELTA..lambda..sub.I is the bandwidth of the incident light. Furthermore, the bandwidth, .DELTA..lambda..sub.cr of an SHG crystal is given by: ##EQU2##
where .lambda. is the wavelength of light and dn/d.lambda. is the derivative cof the refractive index n with respect to wavelength at the appropriate polarization of the fundamental wavelength and second harmonic wavelength, indicated by the subscripts, f and s, respectively.
Thus, the bandwidth of an SHG crystal is a function of the crystal's refractive-index vs. wavelength curve: a fundamental property of the crystal. Furthermore, the bandwidth is inversely proportional to the crystal length. Hence, if one attempts to increase the conversion efficiency by increasing the crystal length, one must also increase the precision of the phase-matching thereby reducing the tolerance for error in the entrance angle of the incoming beam.
Various attempts to improve the efficiency of the SHG process have been and continue to be made. Several researchers have introduced achromatic phase-matching (APM) devices that use angular dispersion so that each wavelength enters the nonlinear crystal at its appropriate phase-matching angle as a way of increasing the bandwidth of the crystal and therefore, increase its efficiency. The crystal and all dispersing optics remain fixed. Because such systems have no moving parts, they are inherently instantaneously tunable, and can be used for nonlinear conversion of tunable or broadband (such as ultrashort) radiation. Most of these devices have used gratings or prisms in combination with lenses which are sensitive to translational misalignment. Also, previous work has considered only the lowest order (linear) term of the media-created dispersion and the phase-matching angle tuning function. Bandwidths of about 10 times the natural bandwidth of the crystal were achieved; larger bandwidths were only obtained by using a divergent beam at the expense of conversion efficiency.
The relationship between the phase-matching angle and the wavelength .lambda. is best approximated by a high order polynomial. By modeling an angularly dispersive optical system such that the dispersion angle(s) of the light propagating through that system, as a function of the wavelength(s), match the phase-matching angle(s) of the SHG crystal, again as a function of wavelength, in both the first and the second order terms of the polynomial, it is possible to bring a much broader band of light wavelengths into the SHGI crystal at the optimum angles for frequency doubling (see FIG. 7). The instant invention seeks to implement this process for increasing the efficiency of frequency doubling through the application of this technique.
The crystal dispersion and phase-matching-angle tuning functions have now been modeled exactly using Sellmeier equations. A "grism" (a prism having a transmission or reflection grating on one surface), which combines the high dispersion property of gratings with the optimum first and second-order behavior of the dispersion angle tuning function has been used in combination with other elements in order to achieve phase-matching over a broader range of incident light wavelengths. Unfortunately, grisms with high diffraction efficiency are not yet available. Indeed, no previous APM device has simultaneously achieved high efficiency and a tuning range greater than approximately 10 times the crystal bandwidth. Furthermore, previous attempts to coalign the otherwise divergent and dispersed second-harmonic beams after passage through the SHG crystal were only partially successful, due in part, to the insufficient precision in matching the second-harmonic dispersion function to the phase-matching function.
An attempt to improve the efficiency of the SHG process is to carry out achromatic phase-matching of the laser pulse incident upon the SHG crystal. FIGS. 1 and 2 show two such conventional approaches. As illustrated, the input light beam 102 is dispersed into its individual frequency components 104 using a diffraction grating 106. As a result, the frequency components 104 of the input light will each propagate at a different angle, with adjacent frequencies having adjacent angles of propagation. Then, using a single lens 108 (FIG. 1) or a two-lens telescope 208 (FIG. 2), these light rays are recombined at the SHG crystal 110. In this manner, all frequencies overlap at the same point and each frequency enters the crystal 110 at its optimal phase-matching angle. Thus, each frequency component of the laser pulse efficiently frequency doubles. In other words, each frequency component essentially acts as an independent and narrowband process, each of which can be quite efficient when a relatively thick crystal is used. Since each frequency component can be treated as a narrowband beam that does not require an SHG crystal with a large bandwidth, a relatively thick crystal can be used.
It is important to note that, because the second harmonic beam produced will be dispersed at an angle, an analogous optical apparatus must be used on the output side of the crystal to reconstruct, i.e., to coalign onto a single path all of the second-harmonic rays/beams within the converted bandwidth.
While these designs potentially achieve improved efficiency in the SHG process itself, they introduce a new inefficiency associated with the diffraction grating. Diffraction gratings are not particularly efficient, and since an additional diffraction grating is required to coalign the second harmonic rays/beams on the other side of the SHG crystal, efficiency is reduced even more. This is especially true if the diffraction grating must operate on ultraviolet light, which will be the most common case in SHG processes. When the inefficiencies of the diffraction gratings 106 are considered, the overall efficiency of the SHG process is reduced by roughly a factor of 4. While the overall efficiency of these designs is still greater than that typically obtainable without achromatic phase-matching using standard crystals, the efficiency is not sufficiently improved that achromatic phase-matching has found practical use.
An alternative approach uses prisms instead of diffraction gratings to disperse the input beam. Both disperse light into its frequency components, but prisms can be anti-reflection-coated or used at Brewster's angle and hence, can result in insignificant loss of efficiency. However, prisms typically have about one tenth the dispersion, which is required for a typical achromatic: phase-matching situation.
As illustrated in FIGS. 3 and 4, in such designs the prisms have been used in conjunction with lens devices to amplify the prism dispersion to appropriate values. In FIG. 3, the input light 301 is incident on a single prism 303 and a two-lens telescope 305 is used to amplify the dispersion of the prism 303 and to focus the light onto the SHG crystal 307. In FIG. 4, light 401 is passed through two appositely oriented prisms 403, 404 and then directed through a single lens 407 to recombine the various frequencies in the SHG crystal 409.
The device depicted in FIG. 3 achieves sufficient dispersion because the two-lens telescope 305 amplifies the dispersion of the prism by 1/M, where M is the magnification of the telescope. A problem associated with such a design is that the group velocity dispersion (the tendency for red wavelengths to travel faster than blue wavelengths) in the system is always positive. Thus, the pulse spreads in time greatly reducing the efficiency of the overall systems for most types of ultrashort light pulses as more fully described below.
The device of FIG. 4 achieves sufficient dispersion because a sufficiently short-focal-length lens 407 can be used to recombine the spatially dispersed rays out of the two-prism assembly to achieve the desired dispersion. While this design can achieve zero (or negative) group-velocity dispersion, it suffers from a different flaw. The angle at which the light rays are incident at the crystal is dependent upon the input position of the input light beam 401 relative to the lens, and in fact, any system which uses lenses is sensitive to the exact position of the lenses.
Finally, as with second-harmonic conversion, other nonlinear optical conversion processes also require angular phase-matching. such processes are mathematically more complex than SHG, but the angles of all input beams must still be precisely controlled to provided efficient phase-matching. Similarly to SHG, the phase-matching angles of all input beams and the resultant angles of all output beams are each a function of all their wavelengths and, in some cases, of the input angles, as well.